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Question
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
Solution
Here,
\[y = \left( \tan^{- 1} x \right)^2 \]
\[\text { Differentiating w . r . t . x, we get }\]
\[ y_1 = \frac{2 \tan^{- 1} x}{1 + x^2}\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[ y_2 = \frac{2 - 4x \tan^{- 1} x}{\left( 1 + x^2 \right)^2}\]
\[ \Rightarrow y_2 = \frac{2}{\left( 1 + x^2 \right)^2} - \frac{2 \tan^{- 1} x \times 2x}{\left( 1 + x^2 \right)^2}\]
\[ \Rightarrow y_2 = \frac{2}{\left( 1 + x^2 \right)^2} - \frac{2x y_1}{\left( 1 + x^2 \right)}\]
\[ \Rightarrow \left( 1 + x^2 \right)^2 y_2 = 2 - 2x\left( 1 + x^2 \right) y_1 \]
\[ \Rightarrow \left( 1 + x^2 \right)^2 y_2 + 2x\left( 1 + x^2 \right) y_1 = 2\]
Hence proved.
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